Quadratic Function Features- Comprehensive Worksheet

What Is a Quadratic Function?

A quadratic function is any function that can be written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is always a parabola.

Parabolas either open upward (like a smile) or downward (like a frown). The shape is consistent, but the position and width change depending on the values of a, b, and c.

If you're working with quadratic functions, you need to understand their key features. This guide breaks down everything you need to know, with practice problems included.

The Three Forms of a Quadratic Function

Quadratic functions appear in three different forms. Each form tells you something different about the parabola.

Standard Form

f(x) = ax² + bx + c

This is the most common form. The coefficient a tells you the direction and width. The constant c is the y-intercept. Finding the vertex from this form requires calculation.

Vertex Form

f(x) = a(x - h)² + k

The vertex is directly visible: it's at (h, k). This form makes graphing much easier. You can see the vertex without doing any math.

Factored Form

f(x) = a(x - r₁)(x - r₂)

The x-intercepts (roots) are visible immediately: they're at r₁ and r₂. This form shows you where the parabola crosses the x-axis.

Key Features of Quadratic Functions

Every quadratic function has these essential features. You need to know how to find and identify each one.

Vertex

The vertex is the highest or lowest point on the parabola. It's the turning point. If a > 0, the vertex is a minimum. If a < 0, the vertex is a maximum.

Finding the vertex from standard form:

Axis of Symmetry

The axis of symmetry is a vertical line that cuts the parabola exactly in half. It passes through the vertex. The equation is always x = -b/(2a).

For the vertex form f(x) = a(x - h)² + k, the axis of symmetry is simply x = h.

Y-Intercept

The y-intercept is where the parabola crosses the y-axis. Set x = 0 and solve. In standard form, the y-intercept is always the constant c.

X-Intercepts (Roots)

The x-intercepts are where the parabola crosses the x-axis. These are the solutions to ax² + bx + c = 0. A parabola can have 0, 1, or 2 x-intercepts.

Use the quadratic formula when factoring doesn't work:

x = (-b ± √(b² - 4ac)) / 2a

The expression under the square root, b² - 4ac, is called the discriminant. It tells you how many x-intercepts exist:

Direction and Width

The coefficient a controls both direction and width:

Comparing the Three Forms

Form Equation Easy to Find Requires Calculation
Standard f(x) = ax² + bx + c Direction, y-intercept Vertex, x-intercepts, axis of symmetry
Vertex f(x) = a(x - h)² + k Vertex, axis of symmetry, direction X-intercepts, y-intercept
Factored f(x) = a(x - r₁)(x - r₂) X-intercepts, direction Vertex, y-intercept, axis of symmetry

How to Analyze a Quadratic Function: Step-by-Step

Here's how to find all the key features of f(x) = 2x² - 12x + 16.

Step 1: Identify the Direction

The coefficient of x² is 2. Since 2 > 0, the parabola opens upward.

Step 2: Find the Y-Intercept

Set x = 0: f(0) = 2(0)² - 12(0) + 16 = 16

The y-intercept is (0, 16).

Step 3: Find the Vertex

x-coordinate: -b/(2a) = -(-12)/(2·2) = 12/4 = 3

y-coordinate: f(3) = 2(9) - 12(3) + 16 = 18 - 36 + 16 = -2

The vertex is (3, -2).

Step 4: Find the Axis of Symmetry

The axis of symmetry is x = 3.

Step 5: Find the X-Intercepts

Set f(x) = 0: 2x² - 12x + 16 = 0

Divide by 2: x² - 6x + 8 = 0

Factor: (x - 2)(x - 4) = 0

X-intercepts are at x = 2 and x = 4. The points are (2, 0) and (4, 0).

Step 6: Check the Discriminant

b² - 4ac = (-12)² - 4(2)(16) = 144 - 128 = 16

Since 16 > 0, there are two real x-intercepts. This matches what we found.

Practice Worksheet: Quadratic Function Features

Work through these problems. Answers are at the end.

Problem 1

For f(x) = x² - 9, identify:

Problem 2

For f(x) = -2(x + 3)² + 5, identify:

Problem 3

For f(x) = 3x² + 6x + 3, find:

Problem 4

How many x-intercepts does f(x) = x² + 4x + 8 have? Use the discriminant to explain your answer.

Problem 5

Write f(x) = x² - 6x + 5 in vertex form by completing the square.

Answers

Problem 1

Problem 2

Problem 3

Problem 4

Discriminant: 4² - 4(1)(8) = 16 - 32 = -16

Since the discriminant is negative, there are no real x-intercepts. The parabola lies entirely above the x-axis.

Problem 5

f(x) = x² - 6x + 5

Take half of -6, square it: (-3)² = 9

Add and subtract 9: f(x) = (x² - 6x + 9) + 5 - 9

Factor: f(x) = (x - 3)² - 4

Vertex form: f(x) = (x - 3)² - 4. The vertex is (3, -4).

Common Mistakes to Avoid

Quick Reference Summary

Feature How to Find It
Direction Look at the sign of a
Vertex (standard form) x = -b/(2a), then find y
Vertex (vertex form) It's (h, k)
Axis of Symmetry x = -b/(2a) or x = h
Y-Intercept Set x = 0 (or look at c)
X-Intercepts Set f(x) = 0, factor or use quadratic formula
Discriminant b² - 4ac tells you how many x-intercepts

That's everything you need to analyze quadratic functions and their features. Work through the practice problems until you can find all features without looking at the steps. The goal is instant recognition, not memorized procedures.